Vacuum Mode Counting and Cosmic Composition: Particle Content and the Magnitude of the Cosmological Constant

This paper is part of a research program proposing that the cosmological constant arises not from total vacuum energy but from the finite-domain residual to which gravity actually couples. This paper develops the fifth component of a research program examining how the particle content of quantum field theory determines the magnitude of the finite-domain vacuum energy residual that acts as the cosmological constant. Papers 3 and 4 of this series showed that the dark energy density satisfies ρΛ = αħc/(lp2L2), where L is the particle horizon, and that the self-consistency condition fixes the ratio R = ρΛ/ρm as a pure number. The dimensionless prefactor α remained undetermined. The present paper motivates and obtains α within the channel-counting framework implied by the UV/IR cancellation structure developed in Paper 2. In this framework, the finite-domain residual vacuum energy arises from quantum field modes below the gravitational collapse scale kG ∼ 1/√(lpL). UV modes cancel under the finite-minus-infinite domain subtraction for all species regardless of mass, while the surviving IR modes contribute a universal geometric energy per fluctuation channel. This yields α = N/(16π2), where N counts the total number of independent quantum field degrees of freedom. The cancellation structure implies that the relevant count is the unsigned sum N = NB + NF, since the boson–fermion sign difference arises from the bulk vacuum contribution removed in the subtraction. For the Standard Model with Dirac neutrinos plus the graviton, N = 126. The magnitude of dark energy is therefore determined at leading order by the particle content of nature together with the gravitational consistency constraint. The resulting prediction gives ΩΛ ≈ 0.685 and Ωm ≈ 0.315, consistent with Planck 2018 at the 1.3% level at leading order, with closer agreement obtained when a small finite-domain boundary correction near the collapse scale is included. The cosmological constant thus emerges as the summed geometric vacuum fluctuation energy of all quantum field degrees of freedom that survive gravitational collapse, behaving as a true cosmological constant with equation of state w = −1. Series Context This paper forms the fifth component of the Finite-Domain Vacuum Energy research program addressing the cosmological constant problem. Papers 1–4 establish the observational constraints, conceptual framework, gravitational consistency conditions, and cosmological self-consistency relations underlying finite-domain vacuum energy. The present paper extends the framework by demonstrating that the prefactor governing the residual vacuum energy density arises from vacuum mode counting across the quantum fields of the Standard Model. The resulting relation α = N/(16π²) links the magnitude of the cosmological constant to the total number of quantum field degrees of freedom, showing that the cosmic composition emerges from the particle content of nature together with the gravitational consistency constraint.